Question: Simplify and expand the following expression: $ \dfrac{4r + 8}{r + 5}+\dfrac{4r}{r + 2} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(r + 5)(r + 2)$ Multiply the first term by $\dfrac{r + 2}{r + 2}$ $ \begin{align*} \dfrac{4r + 8}{r + 5} \times \dfrac{r + 2}{r + 2} & = \dfrac{(4r + 8)(r + 2)}{(r + 5)(r + 2)} \\ & = \dfrac{4r^2 + 16r + 16}{(r + 5)(r + 2)}\end{align*} $ Multiply the second term by $\dfrac{r + 5}{r + 5}$ $ \begin{align*} \dfrac{4r}{r + 2} \times \dfrac{r + 5}{r + 5} & = \dfrac{(4r)(r + 5)}{(r + 2)(r + 5)} \\ & = \dfrac{4r^2 + 20r}{(r + 2)(r + 5)}\end{align*} $ Now we have: $ = \dfrac{4r^2 + 16r + 16}{(r + 5)(r + 2)} + \dfrac{4r^2 + 20r}{(r + 2)(r + 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4r^2 + 16r + 16 + 4r^2 + 20r}{(r + 5)(r + 2)} $ $ = \dfrac{8r^2 + 36r + 16}{(r + 5)(r + 2)}$ Expand the denominator: $ = \dfrac{8r^2 + 36r + 16}{r^2 + 7r + 10}$